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Mathematics > Differential Geometry

Title:
Generalized complex geometry

Abstract: Generalized complex geometry, as developed by Hitchin, contains complex and
symplectic geometry as its extremal special cases. In this thesis, we explore
novel phenomena exhibited by this geometry, such as the natural action of a
B-field. We provide new examples, including some on manifolds admitting no
known complex or symplectic structure. We prove a generalized Darboux theorem
which yields a local normal form for the geometry. We show that there is an
elliptic deformation theory and establish the existence of a Kuranishi moduli
space.
We then define the concept of a generalized Kahler manifold. We prove that
generalized Kahler geometry is equivalent to a bi-Hermitian geometry with
torsion first discovered by physicists. We then use this result to solve an
outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there
exists a Riemannian metric on the complex projective plane which admits exactly
two distinct Hermitian complex structures with equal orientation.
Finally, we introduce the concept of generalized complex submanifold, and
show that such sub-objects correspond to D-branes in the topological A- and
B-models of string theory.